Chapter 4

Linear Algebra Review,

Unitary and Orthogonal

Groups

A significant background in linear algebra will be assumed in later chapters,
and we’ll need a range of specific facts from that subject. These will include
some aspects of linear algebra not emphasized in a typical linear algebra course,
such as the role of the dual space and the consideration of various classes of
invertible matrices as defining a group. For now our vector spaces will be finite
dimensional. Later on we will come to state spaces that are infinite dimensional,
and will address the various issues that this raises at that time.

4.1 Vector spaces and linear maps

A vector spaceV over a fieldkis a set with a consistent way to take linear
combinations of elements with coefficients ink. We will only be using the cases
k=Randk=C, so such finite dimensionalVwill just beRnorCn. Choosing
a basis (set ofnlinearly independent vectors){ej}, an arbitrary vectorv∈V
can be written as
v=v 1 e 1 +v 2 e 2 +···+vnen

giving an explicit identification ofVwithn-tuplesvjof real or complex numbers
which we will usually write as column vectors

v=











v 1

v 2

..

.

vn











The choice of a basis{ej}also allows us to express the action of a linear